Periodic bifurcation from families of periodic solutions for(6)

Let A:D(A)\to E be an infinitesimal generator either of an analytic compact semigroup or of a contractive C_0-semigroup of linear operators acting in a Banach space E. In this paper we give both necessary and sufficient conditions for bifurcation of $T$-pe

Therefore

β(h,ε)= (π2,h(P′(S(h)) I)π2,h) 1(π2,ho(β(h,ε))+π2,hεQ(β(h,ε)+S(h),ε)).DuetoPropertyd)thelastequationimplies(2.4).Wenowproceedtode nethefunctionH.Forthisby(A2)wehavethatr1>0canbetakensu cientlysmallsuchthatS′(h):Rk→E1,hisinvertible.Thuswecande nethefunctionΦξ:Rk→Rk,ξ∈E,asfollows

Φξ(h)=(S′(h)) 1π1,h(ξ S(h)),h∈BRk(h0,r1).

WehavethefollowingpropertiesforΦξ.

1)Φξ0isdi erentiableath0.

2)(Φξ0)(h0)=(S′(h0))

vertiblek×k-matrix.′ 1π1,h0( S′(h0))= I,namely(Φξ0)(h0)isanin-′

Observethatproperty1)isadirectconsequenceofthefactthatξ0 S(h0)=0andthecontinuityofthefunctionh→S 1(h)πh,thereforethedi erentiabilityofπ1,hath=h0isnotnecessaryforthevalidityof1).

Letδ>0besuchthath0istheonlyzeroofΦξ0inBRk(h0,δ).By([10],Theorem6.3)wecanconsiderδ>0su cientlysmallinsuchawaythatd(Φξ0,BRk(h0,δ))=( 1)k.Bythecontinuitypropertyofthetopologicaldegreer1>0canbedi-minished,ifnecessary,insuchawaythatd(Φξ,BRk(h0,δ))=( 1)kforanyξ∈BE(ξ0,r1).Therefore,foranyξ∈BE(ξ0,r1)thereexistsH(ξ)∈BRk(h0,δ)suchthatΦξ(H(ξ))=0.LetusshowthatH(ξ)→h0asξ→ξ0.Indeed,arguingbycontradictionwewouldhaveasequence{ξn}n∈N BE(ξ0,r1),h ∈BRk(h0,δ)suchthatH(ξn)→h =h0asn→∞andthusΦξ0(h )=0contradictingthechoiceofδ>0.Therefore

π1,H(ξ)(ξ S(H(ξ)))=0,ξ∈BE(ξ0,r1).(2.8)

Moreover,weconsiderr2∈(0,r1]su cientlysmalltohave

ξ S(H(ξ)) ≤r1,ξ∈BE(ξ0,r2).(2.9)

Wearenowinthepositiontocompletetheproof.Forthislet(ξ,ε)∈BE(ξ0,r2)×

[0,r2]satisfying(2.1).Then(ξ,ε)alsosatis es

π[P(ξ S(H(ξ))+S(H(ξ))) 1,H(ξ) (ξ S(H(ξ))+S(H(ξ)))+εQ(ξ S(H(ξ))+S(H(ξ)),ε)]=0,

π[P(ξ S(H(ξ))+S(H(ξ))) 2,H(ξ) (ξ S(H(ξ))+S(H(ξ)))+εQ(ξ S(H(ξ))+S(H(ξ)),ε)]=0.

From(2.8),(2.9)andPropertyc)ofLemma2.1wehave

π1,H(ξ)[P(ξ S(H(ξ))+S(H(ξ)))

(ξ S(H(ξ))+S(H(ξ)))+εQ(ξ S(H(ξ))+S(H(ξ)),ε)]=0, β(H(ξ),ε)=ξ S(H(ξ)).

6(2.10)